Sequences and Series
A sequence is an ordered list of numbers following a pattern. A series is the sum of a sequence. Understanding sequences and series is essential for calculus, where limits of infinite series appear in Taylor series, integrals, and fundamental theorems.
In this lesson
1 What Sequences Are
A sequence is a list of numbers in a specific order, typically defined by a rule. The numbers are called terms: a₁ (first term), a₂ (second term), and so on. The general term aₙ gives the nth term as a formula.
Examples: 2, 4, 6, 8, 10... (even numbers). 1, 1, 2, 3, 5, 8, 13... (Fibonacci). 1, 1/2, 1/4, 1/8... (halving each time). Each has a pattern that can be expressed as a formula for aₙ.
2 Arithmetic Sequences
An arithmetic sequence adds the same value (common difference d) each time. General term: aₙ = a₁ + (n−1)d
79Sum of an arithmetic sequence (arithmetic series): Sₙ = n/2 × (a₁ + aₙ) = n/2 × (2a₁ + (n−1)d)
8203 Geometric Sequences
A geometric sequence multiplies by the same value (common ratio r) each time. General term: aₙ = a₁ × rⁿ⁻¹
4,374Sum of finite geometric series: Sₙ = a₁(1−rⁿ)/(1−r) for r ≠ 1.
4 Series and Summation Notation
Summation notation (Σ) compactly represents sums. Σᵢ₌₁ⁿ aᵢ means "sum the terms aᵢ for i from 1 to n."
Σᵢ₌₁⁵ i² = 1² + 2² + 3² + 4² + 5² = 1+4+9+16+25 = 55
Taylor series express functions as infinite polynomial sums: eˣ = 1 + x + x²/2! + x³/3! + ... This allows functions to be approximated to arbitrary precision, which is fundamental to numerical computation and advanced calculus.
5 Infinite Geometric Series
An infinite geometric series converges (sums to a finite value) when |r| < 1: S∞ = a₁/(1−r)
21 + 2 + 4 + 8 + ... has r=2. Since |r| > 1, the series diverges — the sum grows without bound. The formula S∞ = a₁/(1−r) only applies when |r| < 1.
Practice Problems
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